# Investigate Each Limit....(A)

Homework Statement:
Investigate each limit.
Relevant Equations:
See attachment for function.
Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

#### Attachments

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Mentor
Homework Statement:: Investigate each limit.
Relevant Equations:: See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.
Plot some points. What are f(0), f(1/4), f(1/2), f(2), f(2.5), etc.?

Also, you've posted a few threads just now with little or no work shown. That's a violation of forum rules. You have to show some effort. You have the formula for the function -- sketch a graph of it.

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Focus at an interval [n,n+1] where n is an integer. Answer the following questions to help you understand how this function goes
1) What is f(n)
2) What is f(n+1)
3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.

Focus at an interval [n,n+1] where n is an integer. Answer the following questions to help you understand how this function goes
1) What is f(n)
2) What is f(n+1)
3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.
Sorry but I don't get it. Still lost.

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what is f(1) and f(2) equal to for example? Hint: 1 and 2 are integers

nycmathguy
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The definition of the function f(x) tells you that f(x)=1 if x is integer and f(x)=0 if x is not integer.

nycmathguy
The definition of the function f(x) tells you that f(x)=1 if x is integer and f(x)=0 if x is not integer.

I say for (1), the answer is 0.
The answer for (2) is 1.

Yes?

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I say for (1), the answer is 0.
The answer for (2) is 1.

Yes
Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.

Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.
For (1), x tends to an integer. Thus, then f(x) = 1.

For (2), x tends to a rational number. Thus, f(x) = 0.

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Yes but as x tends to an integer, it passes from all sorts of rationals and irrationals (from the left and right of integer) for which f(x)=0.

Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.
What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

Yes?

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If I give you the following definition for f:
f(1)=f(0)=1
f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..

nycmathguy
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What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

Yes?
For both cases the limit is 0. (0 is an integer btw).

Homework Statement:: Investigate each limit.
Relevant Equations:: See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

If I give you the following definition for f:
f(1)=f(0)=1
f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..
Can you elaborate a little more?
It's just not sinking in. In fact, Sullivan stated in his book that this is considered a challenging problem.

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hm ok let me see
If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

nycmathguy
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I think you are confusing the ##\lim_{x\to x_0}f(x)## with the ##f(x_0)##. These two are equal only if the function f is continuous at ##x_0##. But in this problem here we have to deal with a function f that is not continuous at every integer.

nycmathguy
hm ok let me see
If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.
In that case, it is 0.

Delta2
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In that case, it is 0.
Correct now let's say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?

Correct now let's say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?
You said except for x = 0. I say the limit is 1?

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You said except for x = 0. I say the limit is 1?
Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..

nycmathguy
Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..
So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

Delta2
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So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.
That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?

That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?
So, f(x) tends to 0 as x-->0.

Delta2
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So, f(x) tends to 0 as x-->0.
yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .

nycmathguy
yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .
Trust me, I plan to journey through calculus l,ll, and lll. We will see limit questions up the wall.

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Just to check your understanding, if i tell you f(x)=5 for all ##x\neq 0## and f(0)=10, what is the limit of f(x) as x tends to 0?

Just to check your understanding, if i tell you f(x)=5 for all ##x\neq 0## and f(0)=10, what is the limit of f(x) as x tends to 0?
This one is tricky.
I say the limit is 5.

Delta2
Mentor
For (1), x tends to an integer. Thus, then f(x) = 1.
No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.
The question is asking about ##\lim_{x \to 2} f(x)##, not f(x). Even though f(2) = 1, ##\lim_{x \to 2} f(x)## is some other value.
For (1), x tends to an integer. Thus, f(x) = 1.
Again, no.
For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.
First off, 0 is an integer. Second, you're again not distinguishing between function values (e.g. f(0)) and values of the limit. Here the limit expression is ##\lim_{x \to 1/2} f(x)##, which just happens to be the same as f(1/2).
So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.
Most "decimal" numbers are not rational (e.g., ##\pi \approx 3.141592## and ##\sqrt 2 \approx 1.414##), and some rational numbers are integers (e.g., 2/1, 6/2, and so on).

Just to check your understanding, if i tell you f(x)=5 for all x≠0 and f(0)=10, what is the limit of f(x) as x tends to 0?

This one is tricky.
I say the limit is 5.
Right, but it's not tricky if you understand the idea of what a limit means.

Delta2
No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.
The question is asking about ##\lim_{x \to 2} f(x)##, not f(x). Even though f(2) = 1, ##\lim_{x \to 2} f(x)## is some other value.

Again, no.

First off, 0 is an integer. Second, you're again not distinguishing between function values (e.g. f(0)) and values of the limit. Here the limit expression is ##\lim_{x \to 1/2} f(x)##, which just happens to be the same as f(1/2).

Most "decimal" numbers are not rational (e.g., ##\pi \approx 3.141592## and ##\sqrt 2 \approx 1.414##), and some rational numbers are integers (e.g., 2/1, 6/2, and so on).

Right, but it's not tricky if you understand the idea of what a limit means.
Ok. There are many more limits coming our way in time. This is just the beginning of the long journey.

Delta2
Mentor
Ok. There are many more limits coming our way in time.
So make sure you understand the difference between, say, ##f(c)## and ##\lim_{x \to c} f(x)##. For a continuous function f, they will be the same, but not necessarily so for discontinuous or piecewise-defined functions.

nycmathguy
So make sure you understand the difference between, say, ##f(c)## and ##\lim_{x \to c} f(x)##. For a continuous function f, they will be the same, but not necessarily so for discontinuous or piecewise-defined functions.
Will do.

Delta2